Calculus AB: Applications of the Derivative

Problems for "The Mean Value Theorem"

Using the First Derivative to Analyze Functions, page 2

page 1 of 3

First, let's establish some definitions:
f
is said to be increasing on an interval
I
if for all
x
in
I
,
f (x1) < f (x2)
whenever
x1 < x2
.
f
is said to be decreasing on an interval
I
if for all
x
in
I
,
f (x1) > f (x2)
whenever
x1 < x2
. A function is monotonic on an interval
I
if it is only increasing or only decreasing on
I
.

The derivative can help us determine whether a function is increasing or decreasing on an
interval. This knowledge will later allow us to sketch rough graphs of functions.

Let
f
be continuous on
[a, b]
and differentiable on
(a, b)
.
If
f'(x) > 0
for all
x
on
(a, b)
, then
f
is increasing on
[a, b]
.
If
f'(x) < 0
for all
x
on
(a, b)
, then
f
is decreasing on
[a, b]
.

This should make intuitive sense. In the graph below, wherever the slope of the tangent
is positive, the function seems to be increasing. Likewise, wherever the slope of the
tangent is negative, the function seems to be decreasing: